We consider four points $U_1, U_2, U_3, U_4$ in $\mathbb{R}^3$ satisfying $U_1U_2 = U_2U_3 = U_3U_4 = U_4U_1 = 1$, $U_1U_3 = a$ and $U_2U_4 = b$. We set $\Psi(M) = -\frac{1}{2}\Phi(M)$. Show that the vectors $$\left(\begin{array}{r}1\\0\\-1\\0\end{array}\right), \left(\begin{array}{r}0\\1\\0\\-1\end{array}\right), \left(\begin{array}{r}-1\\1\\-1\\1\end{array}\right), \left(\begin{array}{l}1\\1\\1\\1\end{array}\right)$$ form a basis of eigenvectors of the matrix $\Psi(M)$ and determine the eigenvalues of the matrix $\Psi(M)$.
We consider four points $U_1, U_2, U_3, U_4$ in $\mathbb{R}^3$ satisfying $U_1U_2 = U_2U_3 = U_3U_4 = U_4U_1 = 1$, $U_1U_3 = a$ and $U_2U_4 = b$. We set $\Psi(M) = -\frac{1}{2}\Phi(M)$.
Show that the vectors
$$\left(\begin{array}{r}1\\0\\-1\\0\end{array}\right), \left(\begin{array}{r}0\\1\\0\\-1\end{array}\right), \left(\begin{array}{r}-1\\1\\-1\\1\end{array}\right), \left(\begin{array}{l}1\\1\\1\\1\end{array}\right)$$
form a basis of eigenvectors of the matrix $\Psi(M)$ and determine the eigenvalues of the matrix $\Psi(M)$.